Cauchy-Binet for Pseudo-Determinants

@article{Knill2013CauchyBinetFP,
title={Cauchy-Binet for Pseudo-Determinants},
author={Oliver Knill},
journal={arXiv: Rings and Algebras},
year={2013}
}
• O. Knill
• Published 1 June 2013
• Mathematics
• arXiv: Rings and Algebras

Pseudodeterminants and perfect square spanning tree counts

• Mathematics
• 2013
The pseudodeterminant pdet(M) of a square matrix is the last nonzero coefficient in its characteristic polynomial; for a nonsingular matrix, this is just the determinant. If ∂ is a symmetric or

BARYCENTRIC CHARACTERISTIC NUMBERS

• Let
• Mathematics
• 2015
If G is the category of finite simple graphs G = (V,E), the linear space V of valuations on G has a basis given by the f-numbers vk(G) counting complete subgraphs Kk+1 in G. The barycentric

Green functions of Energized complexes

It is proved here that the quadratic energy expression summing over all pairs h(x)^* h(y) of intersecting sets is a signed sum of squares of Green function entries.

A combinatorial expression for the group inverse of symmetric M-matrices

• Mathematics
Special Matrices
• 2021
Abstract By using combinatorial techniques, we obtain an extension of the matrix-tree theorem for general symmetric M-matrices with no restrictions, this means that we do not have to assume the

Gauss-Bonnet for multi-linear valuations

We prove Gauss-Bonnet and Poincare-Hopf formulas for multi-linear valuations on finite simple graphs G=(V,E) and answer affirmatively a conjecture of Gruenbaum from 1970 by constructing higher order

The zeta function for circular graphs

It is proved that the roots of zeta(n,s) converge for n to infinity to the line Re(s)=1/2 in the sense that for every compact subset K in the complement of this line, and large enough n, no root of the zeta function zeta(-s) is in K.

The Kuenneth formula for graphs

The Kuenneth identity is proven using Hodge describing the harmonic forms by the product f g of harmonic forms of G and H and uses a discrete de Rham theorem given by a combinatorial chain homotopy between simplicial and de Ram cohomology.

The Tree-Forest Ratio

. The number of rooted spanning forests divided by the number of spanning rooted trees in a graph G with Kirchhoﬀ matrix K is the spectral quantity τ ( G ) = det(1 + K ) / det( K ) of G by the matrix

A Singular Woodbury and Pseudo-Determinant Matrix Identities and Application to Gaussian Process Regression

• Computer Science, Mathematics
ArXiv
• 2022
An e-cient algorithm and numerical analysis is provided for the presented determinant identities and their advantages in certain conditions which are applicable to computing log-determinant terms in likelihood functions of Gaussian process regression.

References

SHOWING 1-10 OF 69 REFERENCES

On the Coefficients of the Characteristic Polynomial

It is a well-known fact that the first and last non-trivial coefficients of the characteristic polynomial of a linear operator are respectively its trace and its determinant. This work shows how to

The McKean-Singer Formula in Graph Theory

The McKean-Singer formula chi(G) = str(exp(-t L)) which holds for any complex time t, and is proved to allow to find explicit pairs of non-isometric graphs which have isospectral Dirac operators.

Matrix-Forest Theorems

• Mathematics
ArXiv
• 2006
A graph-theoretic interpretation is provided for the adjugate to the Laplacian characteristic matrix for weighted multigraphs and the analogous theorems for (multi)digraphs are established.

The zeta function for circular graphs

It is proved that the roots of zeta(n,s) converge for n to infinity to the line Re(s)=1/2 in the sense that for every compact subset K in the complement of this line, and large enough n, no root of the zeta function zeta(-s) is in K.

The Theory of Matrices

Volume 2: XI. Complex symmetric, skew-symmetric, and orthogonal matrices: 1. Some formulas for complex orthogonal and unitary matrices 2. Polar decomposition of a complex matrix 3. The normal form of

Self-similarity and growth in Birkhoff sums for the golden rotation

• Mathematics
• 2011
We study Birkhoff sums with at the golden mean rotation number with continued fraction pn/qn. The summation of such quantities with logarithmic singularity is motivated by critical KAM phenomena

An Introduction to Linear Algebra

Class notes on vectors, linear combination, basis, span. 1 Vectors Vectors on the plane are ordered pairs of real numbers (a, b) such as (0, 1), (1, 0), (1, 2), (−1, 1). The plane is denoted by R,

Counting rooted forests in a network

We use a recently found generalization of the Cauchy-Binet theorem to give a new proof of the Chebotarev-Shamis forest theorem telling that det(1+L) is the number of rooted spanning forests in a

Trace ideals and their applications

Preliminaries Calkin's theory of operator ideals and symmetrically normed ideals convergence theorems for $\mathcal J_P$ Trace, determinant, and Lidskii's theorem $f(x)g(-i\nabla)$ Fredholm theory